Before you turn this problem in, make sure everything runs as expected. First, restart the kernel (in the menubar, select Kernel$\rightarrow$Restart) and then run all cells (in the menubar, select Cell$\rightarrow$Run All).
Make sure you fill in any place that says YOUR CODE HERE or "YOUR ANSWER HERE", as well as your name and collaborators below:
In [1]:
NAME = "Usama Puri "
COLLABORATORS = "Haithum Haider"
CS110 Fall 2019 - Assignment 1¶
Divide and Conquer Sorting Algorithms¶
This assignment focuses on the implementation of sorting algorithms and analyzing their performance both mathematically (using theoretical arguments on the asymptotic behavior of algorithms ) and experimentally (i.e., running experiments for different input arrays and plotting relevant performance results).
Every CS110 assignment begins with a check-up on your class responsibilities and professional standing, as well as your ability to address one of the course LOs #ComputationalSolutions. Thus to complete the first part of this assignment, you will need to take a screenshot of your CS110 dashboard on Forum where the following is visible: your name. your absences for the course have been set to excused up to session 2.2 (inclusively). This will be evidence that you have submitted acceptable pre-class and make-up work for a CS110 session you may have missed. Check the specific CS110 make-up and pre-class policies in the syllabus of the course.
NOTES:
- Your assignment submission needs to include the following resources:
- A PDF file must be the first resource. This file must be generated from the template notebook where you have written all of the answers (check this link for instructions on how to do this). Make sure that the PDF displays properly (all text and code can be seen within the paper margins).
- Make sure that you submit a neat, clearly presented, and easy-to-read PDF. Please make sure to include page numbers
- Your second resource must be the template notebook you have downloaded from the gist provided and where you included your answers. Submit this file directly following the directions in this picture:
Questions (1)-(7) will be graded on the indicated LOs, please make sure to consult their descriptions and rubrics in the course syllabus. You will not be penalized for not attempting the optional challenge.
After completing the assignment, evaluate the application of the HCs you have identified prior to and while you were working on this assignment and footnote them (refer to these guidelines on how to incorporate HCs in your work). Here are some examples of weak applications of some of the relevant HCs:
- Example 1: “#algorithms: I wrote an implementation of the Bubble sort”.
- This is an extremely superficial use of the HC in a course on Algorithms, and your reference will be graded accordingly. Instead, consider what constitutes an algorithm (see Cormen et al, sections 1.1 and 1.2). Once you have a good definition of an algorithm, think of how this notion helped you approach the implementation of the algorithm, analyze its complexity and understand why it’s important to write an optimal python implementation of the algorithm.
- Example 2: “#dataviz: I plotted nice curves showing the execution time of bubble sort, or I plotted beautiful curves with different colors and labels.”
- Again, these two examples are very superficial uses of the HC #dataviz. Instead consider writing down how do the plots and figures helped you interpret, analyze and write concluding remarks from your experiments. Or write about any insight you included in your work that came from being able to visualize the curves.
- Example 3: “#professionalism: I wrote a nice paper/article that follows all the directions in this assignment.”
- By now, you should realize that this is a poor application of the HC #professionalism. Instead, comment on how you actively considered the HC while deciding on the format, length, and style for writing your report.
- Example 1: “#algorithms: I wrote an implementation of the Bubble sort”.
Your code will be tested for similarity using Turnitin, both to other students’ work and examples available online. As such, be sure to cite all references that you used in devising your solution. Any plagiarism attempts will be referred to the ASC.
Complete the following tasks which will be graded in the designated LOs and foregrounded HCs:
Question 1. [HCs #responsibility and #professionalism; #ComputationalSolutions]¶
Submit a PDF file with a screenshot of your CS110 dashboard with the information described above.
In [2]:
from IPython.display import Image
Image("Screenshot.png")
Out[2]:
Question 2. [#SortingAlgorithms, #PythonProgramming, #CodeReadability]¶
Write a Python 3 implementation of the three-way merge sort discussed in class using the code skeleton below. You should also provide at least three test cases (possibly edge cases) that demonstrate the correctness of your code. Your output must be a sorted Python list.
In [117]:
def merge_sort(lst)-> list:
if len(lst) > 1:
mid = len(lst)//2
list1, list2 = lst[:mid], lst[mid:]
merge_sort(list1)
merge_sort(list2)
i, j, k = 0, 0, 0
while i < len(list1) and j < len(list2):
if list1[i] < list2[j]:
lst[k] = list1[i]
i +=1
else:
lst[k] = list2[j]
j +=1
k +=1
while i < len(list1):
lst[k] = list1[i]
i +=1
k +=1
while j < len(list2):
lst[k] = list2[j]
j +=1
k +=1
return lst
def test_cases():
lst = random.sample(range(100), 20)
return (lst)
testcase1= test_cases()
print('Test Case 1:',testcase1)
print('Sorted List:', merge_sort(testcase1))
testcase2= test_cases()
print('Test Case 2:',testcase2)
print('Sorted List:', merge_sort(testcase2))
testcase3= test_cases()
print('Test Case 3:',testcase3)
print('Sorted List:', merge_sort(testcase3))
Test Case 1: [8, 90, 20, 96, 66, 73, 19, 28, 7, 44, 34, 5, 23, 79, 46, 26, 88, 43, 84, 60] Sorted List: [5, 7, 8, 19, 20, 23, 26, 28, 34, 43, 44, 46, 60, 66, 73, 79, 84, 88, 90, 96] Test Case 1: [20, 90, 21, 79, 28, 89, 61, 94, 84, 74, 2, 14, 41, 69, 51, 87, 33, 0, 11, 46] Sorted List: [0, 2, 11, 14, 20, 21, 28, 33, 41, 46, 51, 61, 69, 74, 79, 84, 87, 89, 90, 94] Test Case 1: [2, 35, 18, 79, 43, 73, 82, 52, 6, 97, 51, 56, 59, 89, 77, 99, 67, 72, 90, 64] Sorted List: [2, 6, 18, 35, 43, 51, 52, 56, 59, 64, 67, 72, 73, 77, 79, 82, 89, 90, 97, 99]
In [150]:
#Python program to implement 3 way merge Sort
def threewaymergesort(x):
if len(x) == 2:
if x[1] < x[0]:
e = x[0]
x[0] = x[1]
x[1] = e
if len(x) > 2:
divide = len(x)// 3
left = x[:divide]
mid = x[divide:divide*2]
right = x[divide*2:]
three_way_mergesort(left)
three_way_mergesort(mid)
three_way_mergesort(right)
a=0
b=0
c=0
d=0
while a < len(left) and c < len(right) and b < len(mid):
if left[a] < mid[b] and left[a] < right[c]:
x[d]= left[a]
a += 1
elif left[a] > mid[b] and right[c]> mid[b]:
x[d] = mid[b]
b += 1
elif left[a] > right[c] and mid[b] > right[c]:
x[d] = right[c]
c += 1
d += 1
while a < len(left) and b < len(mid):
if left[a] < mid[b]:
x[d]=left[a]
a += 1
else:
x[d] = mid[b]
b += 1
d += 1
while a < len(left) and c <len(right):
if left[a] < right[c]:
x[d]= left[a]
a += 1
else:
x[d] = right[c]
c += 1
d += 1
while b < len(mid) and c < len(right):
if mid[b] < right[c]:
x[d] = mid[b]
b += 1
else:
x[d] = right[c]
c += 1
d += 1
while a<len(left):
x[d]= left[a]
a += 1
d += 1
while b<len(mid):
x[d] = mid[b]
b += 1
d += 1
while c<len(right):
x[d] = right[c]
c += 1
d += 1
return(x)
def test_cases():
lst = random.sample(range(100), 10)
return (lst)
testcase1= test_cases()
print('Test Case 1:',testcase1)
print('Sorted List:', threewaymergesort(testcase1))
testcase2= test_cases()
print('Test Case 2:',testcase2)
print('Sorted List:', threewaymergesort(testcase2))
testcase3= test_cases()
print('Test Case 3:',testcase3)
print('Sorted List:', threewaymergesort(testcase3))
Test Case 1: [24, 79, 78, 33, 99, 59, 96, 92, 77, 41] Sorted List: [24, 33, 41, 59, 77, 78, 79, 92, 96, 99] Test Case 2: [54, 35, 66, 60, 63, 24, 26, 49, 23, 33] Sorted List: [23, 24, 26, 33, 35, 49, 54, 60, 63, 66] Test Case 3: [50, 91, 90, 79, 0, 48, 56, 71, 65, 87] Sorted List: [0, 48, 50, 56, 65, 71, 79, 87, 90, 91]
In [88]:
# Please ignore this cell. This cell is for us to implement the tests
# to see if your code works properly.
Question 3. [(#SortingAlgorithms, #PythonProgramming, #CodeReadability, #ComputationalCritique]¶
Implement a second version of a three-way merge sort that calls selection sort when sublists are below a certain length (of your choice) rather than continuing the subdivision process. Justify what might be an appropriate threshold for the input array for applying selection sort.
In [234]:
#Python program to implemen selectionSort if lists are Considerably of a smaller size. But if the size increases a 40 then merge-sort's divide and concur gives better efficiency
#Python program to implement 3 way merge Sort
import random
def selectionSort(A):
for i in range(len(A)):
mini_indx = i
for j in range(i+1, len(A)):
if A[mini_indx] > A[j]:
mini_indx = j
A[i], A[mini_indx] = A[mini_indx], A[i]
def extendedthreewaymergesort(x):
if len(x) == 40:
x = selectionSort(x)
if len(x) > 40:
divide = len(x)// 3
left = x[:divide]
mid = x[divide:divide*2]
right = x[divide*2:]
extendedthreewaymergesort(left)
extendedthreewaymergesort(mid)
extendedthreewaymergesort(right)
a=0
b=0
c=0
d=0
while a < len(left) and c < len(right) and b < len(mid):
if left[a] < mid[b] and left[a] < right[c]:
x[d]= left[a]
a += 1
elif left[a] > mid[b] and right[c]> mid[b]:
x[d] = mid[b]
b += 1
elif left[a] > right[c] and mid[b] > right[c]:
x[d] = right[c]
c += 1
d += 1
while a < len(left) and b < len(mid):
if left[a] < mid[b]:
x[d]=left[a]
a += 1
else:
x[d] = mid[b]
b += 1
d += 1
while a < len(left) and c <len(right):
if left[a] < right[c]:
x[d]= left[a]
a += 1
else:
x[d] = right[c]
c += 1
d += 1
while b < len(mid) and c < len(right):
if mid[b] < right[c]:
x[d] = mid[b]
b += 1
else:
x[d] = right[c]
c += 1
d += 1
while a<len(left):
x[d]= left[a]
a += 1
d += 1
while b<len(mid):
x[d] = mid[b]
b += 1
d += 1
while c<len(right):
x[d] = right[c]
c += 1
d += 1
return(x)
lst = random.sample(range(100), 50)
extendedthreewaymergesort(lst)
def test_cases():
lst = random.sample(range(100), 10)
return (lst)
testcase1= test_cases()
print('Test Case 1:',testcase1)
print('Sorted List:', extendedthreewaymergesort(testcase1))
testcase2= random.sample(range(100), 50)
print('Test Case 2:',testcase2)
print('Sorted List:', extendedthreewaymergesort(testcase2))
testcase3= test_cases()
print('Test Case 3:',testcase3)
print('Sorted List:', extendedthreewaymergesort(testcase3))
Test Case 1: [3, 69, 27, 44, 55, 36, 16, 50, 99, 30] Sorted List: [3, 69, 27, 44, 55, 36, 16, 50, 99, 30] Test Case 2: [79, 16, 2, 15, 61, 32, 69, 55, 60, 40, 93, 20, 41, 91, 39, 42, 7, 25, 76, 75, 11, 97, 74, 21, 80, 65, 86, 99, 54, 82, 18, 10, 35, 36, 95, 73, 8, 78, 19, 47, 23, 29, 87, 85, 94, 81, 49, 68, 52, 30] Sorted List: [7, 25, 35, 36, 76, 75, 11, 79, 16, 2, 15, 61, 32, 69, 55, 60, 40, 93, 20, 41, 91, 39, 42, 95, 73, 8, 78, 19, 47, 23, 29, 87, 85, 94, 81, 49, 68, 52, 30, 97, 74, 21, 80, 65, 86, 99, 54, 82, 18, 10] Test Case 3: [16, 94, 99, 73, 41, 87, 30, 28, 89, 18] Sorted List: [16, 94, 99, 73, 41, 87, 30, 28, 89, 18]
In [309]:
def graph_mergesort_and_selectionsort():
selection_sort_time = []
threeway_mergesort_time = []
for i in range(1,500):
test_array = random.sample(range(-500,500),i)
t1 = time.clock()
selectionSort(test_array)
t2 = time.clock()
threewaymergesort(test_array)
t3 = time.clock()
selection_sort_time.append((t2-t1))
threeway_mergesort_time.append((t3-t2))
#finding the best N
x = range(1,500)
plt.plot(x, selection_sort_time)
plt.plot(x, threeway_mergesort_time)
plt.title('Selection Sort VS Threeway Mergesort')
plt.xlabel('Total Numbers Sorted')
plt.ylabel('Seconds')
plt.legend(['Selection Sort','Threeway Mergesort'], loc = 'upper left')
plt.tight_layout()
plt.show()
graph_mergesort_and_selectionsort()
In [308]:
# Please ignore this cell. This cell is for us to implement the tests
# to see if your code works properly.
Question 4 [#SortingAlgorithms, #PythonProgramming, #CodeReadability]¶
Bucket sort (or Bin sort) is an algorithm that takes as inputs an $n$-element array and the number of buckets, $k$, to be used during sorting. Then, the algorithm distributes the elements of the input array into $k$-different buckets and proceeds to sort the individual buckets. Then, merges the sorted buckets to obtained the sorted array. Here is pseudocode for the BucketSort algorithm:
The BucketSort above calls the function GetBucketNum (see the pseudocode below) to distribute all the elements of array $A$ into $k$-buckets. Every element in the array is assigned a bucket number based on its value (positive or negative numbers). GetBucketNum returns the bucket number that corresponds to element $A[i]$. It takes as inputs the element of the array, $A[i]$, the max and min elements in $A$, the size of the intervals in every bucket (e.g., if you have numbers with values between 0 and 100 numbers and 5 buckets, every bucket has an interval of size $20 = [100-0]/5$). Notice that in pseudocode the indices of the arrays are from 1 to $n$. Thus, GetBucketNum consistently returns a number between 1 and $n$ (make sure you account for this in your Python program).
Write a Python 3 implementation of BucketSort that uses the selection sort algorithm for sorting the individual buckets in line 10 of the algorithm.
In [324]:
from math import ceil
import random
def selectionSort(A):
for i in range(len(A)):
mini_indx = i
for j in range(i+1, len(A)):
if A[mini_indx] > A[j]:
mini_indx = j
A[i], A[mini_indx] = A[mini_indx], A[i]
def bucketSort(A):
k=len(A)
if k==0:
return A
minimum=min(A)
maximum=max(A)
results = []
size=ceil(((maximum-minimum)/k))
buckets=[[] for _ in range(k)]
for i in range(k):
b = getbucketnum(A[i],minimum, maximum, size, k)
buckets[b].append(A[i])
for i in range(k):
selectionSort(buckets[i])
for i in range(k):
results+=buckets[i]
return results
def getbucketnum(a, minimum, maximum, size, k):
if a==maximum:
j=k
elif a==minimum:
j=1
else:
j=1
while a > minimum+ (size*j):
j+=1
return j-1
def test_cases():
lst = random.sample(range(100), 10)
return (lst)
testcase1= test_cases()
print('Test Case 1:',testcase1)
print('Sorted List:', bucketSort(testcase1))
testcase2= test_cases()
print('Test Case 2:',testcase2)
print('Sorted List:', bucketSort(testcase2))
testcase3= test_cases()
print('Test Case 3:',testcase3)
print('Sorted List:', bucketSort(testcase3))
Test Case 1: [64, 29, 30, 10, 60, 14, 94, 93, 20, 86] Sorted List: [10, 14, 20, 29, 30, 60, 64, 86, 93, 94] Test Case 2: [81, 41, 74, 80, 40, 32, 7, 47, 18, 6] Sorted List: [6, 7, 18, 32, 40, 41, 47, 74, 80, 81] Test Case 3: [77, 81, 61, 0, 10, 65, 51, 54, 19, 56] Sorted List: [0, 10, 19, 51, 54, 56, 61, 65, 77, 81]
In [92]:
# Please ignore this cell. This cell is for us to implement the tests
# to see if your code works properly.
Question 5 [#SortingAlgorithms, #PythonProgramming, #CodeReadability]¶
Implement a second version of the BucketSort algorithm. This time in line 10 of BucketSort use the Bucket sort recursively until the size of the bucket is less than or equal to k, the base case for the recursion.
In [157]:
from math import ceil
import random
def extendedBucketSort(A):
k=len(A)
minimum=min(A)
maximum=max(A)
results = []
size=ceil((maximum-minimum)/k)
buckets=[[] for _ in range(k)]
for i in range(k):
b = getbucketnum(A[i],minimum, maximum, size, k)
buckets[b].append(A[i])
for i in range(k):
bucketSort(buckets[i])
for i in range(k):
results+=buckets[i]
return results
def getbucketnum(a, minimum, maximum, size, k):
if a==maximum:
j=k
elif a==minimum:
j=1
else:
j=1
while a > minimum+ (size*j):
j+=1
return j-1
def test_cases():
lst = random.sample(range(100), 10)
return (lst)
testcase1= test_cases()
print('Test Case 1:',testcase1)
print('Sorted List:', extendedBucketSort(testcase1))
testcase2= test_cases()
print('Test Case 2:',testcase2)
print('Sorted List:', extendedBucketSort(testcase2))
testcase3= test_cases()
print('Test Case 3:',testcase3)
print('Sorted List:', extendedBucketSort(testcase3))
Test Case 1: [0, 50, 49, 91, 32, 68, 80, 6, 10, 2] Sorted List: [0, 6, 10, 2, 32, 50, 49, 68, 80, 91] Test Case 2: [94, 48, 8, 29, 79, 93, 55, 65, 9, 92] Sorted List: [8, 9, 29, 48, 55, 65, 79, 94, 93, 92] Test Case 3: [85, 82, 60, 54, 42, 76, 77, 84, 80, 23] Sorted List: [23, 42, 54, 60, 76, 77, 82, 84, 80, 85]
In [94]:
# Please ignore this cell. This cell is for us to implement the tests
# to see if your code works properly.
Question 6 [#ComplexityAnalysis, #ComputationalCritique]¶
Analyze and compare the practical run times of regular merge sort (i.e., two-way merge sort), three-way merge sort, and the extended merge sort from (3) by producing a plot that illustrates how every running time and number of steps grows with input size. Make sure to:
- define what each algorithm's complexity is
- enumerate the explicit assumptions made to assess each run time of the algorithm's run time.
- and compare your benchmarks with the theoretical result we have discussed in class.
In [95]:
def graph_mergesort():
regular_mergesort_time = []
threeway_mergesort_time = []
extended_mergesort_time = []
for i in range(1,5000,20):
t_array = random.sample(range(-5000,5000),i)
t1 = time.clock()
merge_sort(t_array)
t2 = time.clock()
threewaymergesort(t_array)
t3 = time.clock()
extendedthreewaymergesort(t_array)
t4 = time.clock()
regular_mergesort_time.append(t2-t1)
threeway_mergesort_time.append(t3-t2)
extended_mergesort_time.append(t4-t3)
x = range(1,5000,20)
plt.plot(x, regular_mergesort_time)
plt.plot(x, threeway_mergesort_time)
plt.plot(x, extended_mergesort_time)
plt.title('Difference between Runtime of Different Mergesort Techniques')
plt.xlabel('Total Numbers Sorted')
plt.ylabel('Seconds')
plt.legend(['Merge Sort','Threeway Merge Sort','Extendend Merge Sort'], loc = 'upper left')
plt.tight_layout()
plt.show()
graph_mergesort()
[-2026] [-2026] [-4921, -4173, -3728, -2815, -2652, -2228, -2202, -1816, -1293, -918, -440, -224, -197, 981, 2379, 2763, 3723, 3955, 4012, 4099, 4730] [-4921, -4173, -3728, -2815, -2652, -2228, -2202, -1816, -1293, -918, -440, -224, -197, 981, 2379, 2763, 3723, 3955, 4012, 4099, 4730] [-4947, -4627, -4509, -4268, -4053, -3725, -3678, -3514, -3156, -2400, -2381, -1948, -1761, -1309, -1100, -930, -912, -617, -537, -457, -305, -205, 153, 183, 727, 863, 1160, 1728, 1738, 2266, 2339, 2635, 2665, 3266, 3567, 3661, 3830, 4352, 4409, 4631, 4765] [-4947, -4627, -4509, -4268, -4053, -3725, -3678, -3514, -3156, -2400, -2381, -1948, -1761, -1309, -1100, -930, -912, -617, -537, -457, -305, -205, 153, 183, 727, 863, 1160, 1728, 1738, 2266, 2339, 2635, 2665, 3266, 3567, 3661, 3830, 4352, 4409, 4631, 4765] [-4957, -4628, -4394, -4159, -4128, -4080, -4059, -3938, -3857, -3783, -3741, -3632, -3535, -3425, -2980, -2933, -2881, -2845, -2547, -2290, -2095, -1748, -1613, -1223, -1016, -994, -991, -899, 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Question 7. [#ComplexityAnalysis, #ComputationalCritique]¶
Analyze and compare the practical run times of regular merge sort (i.e., two-way merge sort), Bucket sort and recursive sort from (5) by producing a plot that illustrates how each running time grows with input size. Make sure to:
- define what each algorithm's complexity is
- enumerate the explicit assumptions made to assess each run time of the algorithm's run time.
- and compare your benchmarks with the theoretical result we have discussed in class.
In [ ]:
def graph_mergesort_and_bucket_sort():
regular_mergesort_time = []
regular_bucketsort_time = []
recursive_bucketsort_time = []
for i in range(1,5000,20):
t_array = random.sample(range(-5000,5000),i)
t1 = time.clock()
merge_sort(t_array)
t2 = time.clock()
bucketSort(t_array)
t3 = time.clock()
extendedBucketSort(t_array)
t4 = time.clock()
regular_mergesort_time.append(t2-t1)
regular_bucketsort_time.append(t3-t2)
recursive_bucketsort_time.append(t4-t3)
x = range(1,5000,20)
plt.plot(x, regular_mergesort_time)
plt.plot(x, regular_bucketsort_time)
plt.plot(x, recursive_bucketsort_time)
plt.title('Difference between Runtime of Different Sorting Techniques')
plt.xlabel('Total Numbers Sorted')
plt.ylabel('Seconds')
plt.legend(['Merge Sort','Bucket Sort','Recursive Bucket Sort'], loc = 'upper left')
plt.tight_layout()
plt.show()
graph_mergesort_and_bucket_sort()
[Optional challenge] Question 8 (#SortingAlgorithm and/or #ComputationalCritique)¶
Implement k-way merge sort, where the user specifies k. Develop and run experiments to support a hypothesis about the “best” value of k.